Density, must

As the number of atoms in the asyimnetric unit increases, the solution of a structure by any of these phase-independent methods becomes more difficult, and by 1950 a PhD thesis could be based on a single crystal structure. At about that time, however, several groups observed that the fact that the electron density must be non-negative everywhere could be exploited to place restrictions on possible phases. The first use of this fact was by D Marker and J S Kasper [24], but their relations were special cases of more general relations introduced by J Karle and H Hauptman [25]. Denoting by A. the set of indices h., k., /., the Karle-Hauptman condition states that all matrices of the fonu... 

In order that the flowmeter may be used during an ozoni-zation when tube J is empty it is necessary to mark the heights of liquid on both tubes. The same liquid head (height X density) must be present in the absorption tubes during calibra-... 

Gauss s theorem, in mathematics, says a change in a volume s density must be accompanied by flow through the boundary. Leak path analysis is a qualitative interpretation of Gauss s theorem. 

The fluid density must be for the combined stream of oil and gas and should be calculated at the average gas temperature. 

For gases other than air the density must be calculated because the curves of Figure 12-148 are for 0.075 Ib/fti density air. If the gas handled has a density at 70°F and... 

The significance of the Flade potential Ef, passivation potential pp, critical current density /pn, passive current density, etc. have been considered in some detail in Sections 1.4 and 1.5 and will not therefore be considered in the present section. It is sufficient to note that in order to produce passivation (a) the critical current density must be exceeded and b) the potential must then be maintained in the passive region and not allowed to fall into the active region or rise into the transpassive region. It follows that although a high current density may be required to cause passivation ) only a small current density is required to maintain it, and that in the passive region the corrosion rate corresponds to the passive current density (/p, ). 

Chromium plating from hexavalent baths is carried out with insoluble lead-lead peroxide anodes, since chromium anodes would be insoluble (passive). There are three main anode reactions oxidation of water, reoxidation of Cr ions (or more probably complex polychromate compounds) produced at the cathode and gradual thickening of the PbOj film. The anode current density must balance the reduction and reoxidation of trivalent chromium so that the concentration reaches a steady state. From time to time the PbOj film is removed as it increases electrical resistance. 

Its experimental confirmation provides information about the free rotation time tj. However, this is very difficult to do in the Debye case. From one side the density must be high enough to reach the perturbation theory (rotational diffusion) region where i rotational relaxation which is valid at k < 1. The two conditions are mutually contradictory. The validity condition of perturbation theory... 

It was underlined twice in [251] that the use of fitting laws ties in with the validity of the impact theory, and the present results have to be interpreted cautiously . At high pressure nonlinear increase of rates of energy relaxation and vibrational dephasing with gas density must be... 

The surface of the base metal is anodically polarized under the effect of local cells. For a graphical analysis of the phenomena, one must construct the polarization curves for the partial currents at the base metal as well as the overall anodic 4 vs. E curve reflecting the effective rate of dissolution of this metal under anodic polarization. The rate of the cathodic process, 4, at the inclusions is described by the corresponding cathodic polarization curve (since the surface areas of anodic and cathodic segments differ substantially, currents rather than current densities must be employed here). At open circuit the two rates are identical. 

The incorporation of phosphorus yields fourfold-coordinated P atoms, which are positively charged, as phosphorus normally is threefold coordinated. This substitutional doping mechanism was described by Street [52], thereby resolving the apparent discrepancy with the so-called S N rule, with N the number of valence electrons, as originally proposed by Mott [53]. In addition, the incorporation mechanism, because charge neutrality must be preserved, leads to the formation of deep defects (dangling bonds). This increase in defect density as a result of doping explains the fact that a-Si H photovoltaic devices are not simple p-n diodes (as with crystalline materials) an intrinsic layer, with low defect density, must be introduced between the p- and n-doped layers. 

This confirms the earlier interpretation that the exponent reflects the entropy of the reservoirs only, and that the contribution from internal changes of the subsystem has been correctly removed. During the adiabatic transition the reservoirs do not change, and so the probability density must be constant. Obviously there is an upper limit on the time interval over which this result holds since the assumption that X Xs implies that A( (x)x -C XS. ... 

An ideal ion source must possess high brightness, and must produce an ion beam of homogeneous composition with a small energy spread. The ion current density must be easily monitored and remain constant across the beam cross-section. 

The conclusion is then that the wavefunction representing a system composed of indistinguishable particles must be either symmetric or antisymmetric under the permutation operation. On purely physical grounds, this result is apparent, as the probability density must be independent of the permutation of indistinguishable particles or 1 (1,2) 2 = (2,1) 2. 

The many-electron wave function (40 of any system is a function of the spatial coordinates of all the electrons and of their spins. The two possible values of the spin angular momentum of an electron—spin up and spin down—are described respectively by two spin functions denoted as a(co) and P(co), where co is a spin degree of freedom or spin coordinate . All electrons are identical and therefore indistinguishable from one another. It follows that the interchange of the positions and the spins (spin coordinates) of any two electrons in a system must leave the observable properties of the system unchanged. In particular, the electron density must remain unchanged. In other words, 4 2 must not be altered... 

The ionic densities must in turn depend on the potential x). We choose (j) po) = 0 as our reference, and apply Boltzmann statistics ... 

By the ideal gas law, assuming no change in molecular weight, p = pRT/M, or density must also be constant. Consequently, Equation (10.68) becomes... 

In the VEP, currents used are between 600 and 1200 A at potentials between 30 and 60 V. The vibration frequency of the wire electrode is up to 500 Hz. The materials atomized via VEP include mild steel, Cr-Ni steel, Cu-Ni alloy and tungsten. The VEP is carried out in an inert atmosphere (typically argon) for most alloys, but the arc is struck under water for tungsten wire. Wire diameter is 1-4 mm, and its feed rate is 1.7-4.3 m/min. The feed rate and current density must be determined properly according to the relationship between these two variables. At lower current densities, the wire electrode tends to stick to the rotating electrode. At higher current densities, the wire electrode becomes overheated, causing it to bend or even rupture. 

Examples of p(r) are energy density, charge density, current density (see Section 4.6), difference density (difference between a final density and an initial density), electric moment density, magnetic moment density, local reactivity functions (see Section 4.5.2), force density, etc. Note that, for ensuring the stability of matter, the net force density must vanish everywhere in space. The concept of a PDF has generated many significant developments in interpretative quantum chemistry. 

If population analysis is not synonymous with the concept of an AIM, it becomes necessary to introduce a proper set of requirements before one can speak of an AIM. An AIM is a quantum object and as such has an electron density of its own. This atomic electron density must obviously be positive definite and the sum of these atomic densities must equal the molecular density. Each atomic density pA(r) can be obtained from the molecular density p(r) in the following way ... 

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